cos (2x) = cos x
2 cos^2 x -1 = cos x using the double angle formula
2 cos ^2 x -cos x -1 =0
factor
(2 cos x+1) ( cos x -1) = 0
using the zero product property
2 cos x+1 =0 cos x -1 =0
2 cos x = -1 cos x =1
cos x = -1/2 cos x=1
taking the arccos of each side
arccos cos x = arccos (-1/2) arccos cos x = arccos 1
x = 120 degrees x=-120 degrees x=0
remember you get 2 values ( 2nd and 3rd quadrant)
these are the principal values
now we need to add 360
x = 120+ 360n x=-120+ 360n x = 0 + 360n where n is an integer
P(8 or 12) = 1/6
P(not 8 or 12 = 5/6
Let the expected points fro one roll be X.

The answer is -(1/6) points.
The values of x and y that will make LMNO a parallelogram are: x = 9, y = 1.5.
<h3>
Properties of a Parallelogram</h3>
- The opposite sides of a parallelogram are parallel to each other and are also of equal lengths.
- Diagonals of a parallelogram are congruent to each other and also bisect each other into equal halves.
Therefore:
ON = LM
Substitute
7x - 5 = 6x + 4
Add like terms
7x - 6x = 5 + 4
x = 9
OL = NM
Substitute
8y + 3 = x + 6
Plug in the value of x
8y + 3 = 9 + 6
8y + 3 = 15
8y = 15 - 3
8y = 12
y = 12/8
y = 1.5
Therefore, the values of x and y that will make LMNO a parallelogram are: x = 9, y = 1.5.
Learn more about parallelogram on:
brainly.com/question/7200842
This function would have a maximum.
Since we are subtracting by a -4 for each increase in x, we know that the numbers will continue to go down. Given this fact, we know the number will never be higher than when we started, but the number could go infinitely low. As a result we have a maximum and no minimum.